Long-term error analysis of low-regularity integrators for stochastic Schr$\ddot{\rm o}$dinger equations

TL;DR

The paper addresses long-time integration of the cubic stochastic Schrödinger equation with additive noise by introducing an explicit non-resonant low-regularity stochastic integrator based on a twisted-variable transformation and the regularity compensation oscillation (RCO) technique. It proves strong convergence of the method with rate $\tau^{\min(\tfrac{1}{2},(\nu-1)/2,\gamma)}$ in $L^{2p}(\Omega; H^{\sigma})$ and establishes a long-time $H^1$ error bound of $O(\tau^{1/2}\varepsilon^{\min(2,(q-2))})$ up to time $O(\varepsilon^{-2})$ for small data $u^0$ and noise size $\varepsilon^q$ with $q>2$. The numerical results corroborate the theory, showing improved long-time stability compared to existing integrators. These findings enhance reliable long-time simulations of stochastic nonlinear Schrödinger dynamics under low regularity.

Abstract

In this paper, we design an explicit non-resonant low-regularity integrator for the cubic nonlinear stochastic Schr$\ddot{\rm o}$dinger equation (SNLSE) with the aim of allowing long time simulations. First, we carry out a strong error analysis for the new integrator. Next we provide, for small initial data of size $\mathcal{O}(\varepsilon), \varepsilon \in (0,1]$, and noise of size $\varepsilon^q, \ q>2$, long-term error estimates in the space $L^{2p}(Ω, H^1), p\ge 1$, revealing an error of size $\mathcal{O}\left(τ^{\frac{1}{2}}\cdot \varepsilon^{{\rm min}(2,(q-2))}\right)$ up to time $\mathcal{O}(\varepsilon^{-2})$. This is achieved with the regularity compensation oscillation technique \cite{sch0,bao}, which has been here introduced and exploited for the stochastic setting. A numerical experiment confirms the superior long-time behaviour of our new scheme, compared to other existing integrators.

Figures (1)

• Figure 1: Long term squared $L^2(\Omega,H^1)$ error comparison for methods SNRLI1 \ref{['met']} and method \ref{['met_ref1']}, applied to equation \ref{['sch_w']} on $[0,100]$, with $\varepsilon=0.1$ and $q=3.5$, with $\tau=0.01$. Here, we have employed $M=10^2$ paths. $u^0(x)=\frac{2}{2-\cos(x)}, \ x\in [-\pi,\pi]$.

Theorems & Definitions (20)

• Lemma 2.1
• Proposition 2.1
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.1
• proof